The Theorems
1w. A finite skew field is a field
[Wedderburn]. 1a-z. A finite alternative division ring is a field
[Artin-Zorn]. 2. A finite integral domain is a field [pigeon hole principle] 3. For each prime number p and positive integer n there exists a
(up to isomorphism) unique finite field of order q = pn, a Galois Field,
namely, the splitting field of the polynomial gq(X) = Xq - X
over Zp, to be denoted by GF(pn) or Fq.
Moreover Fq is a linear space of dimension n over Zp
and can be constructed by adjunction of a root a, say, whose (irreducible) minimal polynomial
over Zp has degree n.
The group of all automorphisms of Fq, called the Galois group, Gal,
of Fq (over Zp) is cyclic of order n and generated by
the Frobenius
morphism f defined by f(x)=xp.
[Formally the field Fq may be constructed from the principal ideal ring
Zp[X]of polynomials (in X) over Zp by dividing out
the maximal ideal generated by the minimal polynomial of a.
Mapping the residue class of X onto a will produce a natural isomorphism onto the classical
(more intuitive and more practical) model of Fq above.] 4. The multiplicative group of Fq is cyclic (of order q-1).
A generator of it is called a primitive element of the field. 5. Now consider the disjoint cycle decomposition of f viewed as a permutation of the elements of
Fq. By grouping the elements of Fq according to these cycles,
that is, according to their orbits under Gal, which by the orbit theorem have lenghts dividing the order
n of Gal, the factorization of gq(X) over Fq: gq(X) = the product, over t in
Fq, of all monic linear polynomials (X-t)
can be transformed into the prime factorization over Zp: gq(X) = the product of
all monic irreducible polynomials over Zp whose degree k, say,
divide n. Conversely, if b is an element of Fq and
K is a minimal subfield containing b with order pk, say, then n = km,
where m is the dimension of Fq over K, and the minimal polynomial of
b over Zp factorizes over Fq as a product of
k linear polynomials of the form:
(X-b)(X-fm(b))...(X-fm(k-1)(b))
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![[5, a Desargues' Configuration]](./Desargues.jpg){kind=link}
